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Properties

Label 2-275-11.4-c1-0-11
Degree $2$
Conductor $275$
Sign $0.901 - 0.431i$
Analytic cond. $2.19588$
Root an. cond. $1.48185$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.04 + 1.48i)2-s + (0.762 − 2.34i)3-s + (1.35 + 4.15i)4-s + (5.03 − 3.66i)6-s + (−0.646 − 1.99i)7-s + (−1.85 + 5.69i)8-s + (−2.49 − 1.81i)9-s + (−1.64 + 2.87i)11-s + 10.7·12-s + (1.04 + 0.757i)13-s + (1.63 − 5.02i)14-s + (−5.16 + 3.74i)16-s + (−2.41 + 1.75i)17-s + (−2.40 − 7.40i)18-s + (0.664 − 2.04i)19-s + ⋯
L(s)  = 1  + (1.44 + 1.04i)2-s + (0.440 − 1.35i)3-s + (0.675 + 2.07i)4-s + (2.05 − 1.49i)6-s + (−0.244 − 0.752i)7-s + (−0.654 + 2.01i)8-s + (−0.831 − 0.604i)9-s + (−0.496 + 0.867i)11-s + 3.11·12-s + (0.289 + 0.210i)13-s + (0.436 − 1.34i)14-s + (−1.29 + 0.937i)16-s + (−0.586 + 0.426i)17-s + (−0.567 − 1.74i)18-s + (0.152 − 0.469i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.901 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.901 - 0.431i$
Analytic conductor: \(2.19588\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1/2),\ 0.901 - 0.431i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.75348 + 0.625296i\)
\(L(\frac12)\) \(\approx\) \(2.75348 + 0.625296i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 + (1.64 - 2.87i)T \)
good2 \( 1 + (-2.04 - 1.48i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.762 + 2.34i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (0.646 + 1.99i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.04 - 0.757i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.41 - 1.75i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.664 + 2.04i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 8.77T + 23T^{2} \)
29 \( 1 + (0.189 + 0.582i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.94 - 2.14i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.578 - 1.77i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.57 - 4.85i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 5.17T + 43T^{2} \)
47 \( 1 + (-2.25 + 6.94i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.38 + 1.72i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.00 + 6.17i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.406 + 0.295i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 7.80T + 67T^{2} \)
71 \( 1 + (-9.14 + 6.64i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.43 - 10.5i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.33 + 3.14i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-8.77 + 6.37i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 4.32T + 89T^{2} \)
97 \( 1 + (0.284 + 0.206i)T + (29.9 + 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.47330209424360188757285612956, −11.65639413929561690478691555874, −10.10026180617752895888393429963, −8.350392887778047252903365076062, −7.68095524212594706210190753689, −6.83793331822232685191352886946, −6.29235240437952840880110684673, −4.84356144438638546313830698361, −3.70905877981569747212485563761, −2.21276154591427977871533238375, 2.44022670697732174534494441517, 3.41441990340450517545204177641, 4.27345188506925498416417711463, 5.36791075723499815294665105798, 6.11838527110952569637642529481, 8.286573179832769366363724964683, 9.405713971997640018824870031902, 10.22560820624665147734025464407, 10.95117530896154536465099382688, 11.82529162305878919575820018259

Graph of the $Z$-function along the critical line